Introduction to the Theory of Random Processes
The author, one of the leading researchers in the field of stochastic control, presents an introductory and, at the same time, advanced text aimed at undergraduate students majoring in probability and statistics. The theory of random processes, an extremely vast and complex part of mathematics, is laid out gradually and in a manner that respects the student's mathematical background; the presentation is rigorous and, importantly, allows him/her to appreciate the true probabilistic flavour of the formal theory. The main topics of the book are Wiener process, stationary processes, infinitely divisible processes and the Itô stochastic calculus that includes the strong theory of stochastic differential equations. There are some methodological novelties to be found in the text that deserve to be mentioned: The Itô stochastic integral which is introduced at a very early stage (Chapter 2) is viewed as a particular case of the integral with a random orthogonal measure as the integrator. Stochastic integration in this generality is extensively used. The spectral representation of trajectories of stationary processes and a representation of trajectories of infinitely divisible processes through jump measures are examples. About 130 exercises (hints available) accompany the bulk of mathematical reasoning, some of them being used in the main text. The reviewer's opinion is that the book meets the advertised purposes and may well serve as an introduction to the modern theory of stochastic processes.